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We need to wrap up this cube-shaped present, remembering that we
can have no overlaps. What shapes can you find to use?

Painted Faces

Stage: 2 Challenge Level:

Painted Faces

This activity is all about imagining, as you might do when you listen to a story or poem.

There is no need to write or draw anything

BUT

talking about it could be good

SO

find a friend to do this with!

Imagine a 3 by 3 by 3 cube made from 27 smaller cubes.
Perhaps it's here - but behind this cloud!

So, remove the cloud in your mind and there it is!

It's hanging in the air in front of you so you can just see one face of it and that face has been painted red

The right-hand side face has been painted yellow

The left-hand side face has been painted blue

The top face has been painted white

The underneath bottom face has been painted black

The back face has been painted green.

1. How many small cubes have just two of their faces painted?
2. Where are they?
3. What are the two colours on each of those small cubes that have two faces painted?

Now you could try the same things with a larger starting cube, that is 4 by 4 by 4, and answer the same three questions.

If you'd like to take these ideas a bit further, have a look at Painted Cube .

Why do this problem?

This might be a good starting exercise, if, as a teacher, you've never done any similar imaginative thinking (visualising) in Maths. You may find it best to introduce these kinds of ideas in small groups to start with, before having the whole class embark on such an adventure.

Possible approach

It can be very hard to prevent yourself from saying too much when the pupils are explaining what is going on in their minds. Questions such as "Tell me about what you are imagining" and "Tell me about how you're counting" are probably useful "open" questions to use. Try to avoid halting a pupil's thinking by saying "Yes I know exactly what you mean" too soon! A natural progression from the
problem is to look at the other "kinds" of small cubes which make up the larger cube, for example those with just one face painted, three faces painted, four faces painted etc. and of course no faces painted.

When going on to the larger 4 by 4 by 4 cube, it would be worth asking pupils to predict the numbers before thinking too much.

Key questions

Possible extension

If you are keen to focus on generalisations for still larger cubes, you could look at Painted Cube .

The NRICH Project aims to enrich the mathematical experiences of all learners. To support this aim, members of the
NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to
embed rich mathematical tasks into everyday classroom practice. More information on many of our other activities
can be found here.